Distributed reflector in a microring resonator

ABSTRACT

A component having a microring resonator and grating, coupled to a waveguide is described. By selection of the grating period, and other parameters such as the grating-waveguide coupling coefficient, an efficient filter may be designed and constructed. The component may be used in passive devices such as add-drop multiplexers or sensors, or in active devices such as lasers. Designs having essentially no response sidelobes, very narrow effective bandwidths, and fast filter roll-offs may permit compact devices to be produced, when compared with typical distributed sampled Bragg grating structures.

This application claims the benefit of U.S. provisional application61/382,537, filed on Sep. 14, 2010, which is incorporated herein byreference.

TECHNICAL FIELD

The present application may be related to the design and use of gratingsin optical components and systems.

BACKGROUND

A microring resonator (MRR) has been used in various photonic devicesdue to the potentially large quality factor (Q) and small size of theMRR. Such devices have been used as the building blocks of integratedoptical filters, switches, modulators, logic gates, memories, delaylines, sensors, and laser cavities.

The distributed Bragg reflector (DBR) is a technology that has enablednumerous optoelectronic devices including vertical cavity lasers, narrowlinewidth lasers, fiber Bragg gratings, dichroic mirrors, and dispersioncompensation filters. The DBR laser has many applications includingcommunications, metrology, sensing, and spectroscopy.

A dispersive photonic bandgap crystal (PhC) in an MRR have beeninvestigated for narrowing a resonator linewidth using enhanced modaldispersion near the PhC bandgap. The device was designed to have a sharpnotch in reflection at resonance by coupling a PhC MRR to a broadbandexternal linear reflective PhC in a bus waveguide. Good notch filteringperformance requires that the external PhC be both highly reflective andhave minimal scattering loss, and also requires precise phase matchingof the PhC k-vector to the MRR. These constraints make fabrication ofthe device difficult. Also, when the PhC is very highly reflective, thestrength of the field in the ring is approximately less than or equal tothe input field strength in the bus waveguide.

SUMMARY

A family of microring devices, which may be referred to as a distributedreflector in an optical ring raveguide is disclosed. An example of sucha device is a distributed Bragg grating in a microring resonator(DBR-MRR). The buildup of field strength in a ring resonatorconfiguration yields multiple partial reflection encounters with a sameset of grating elements. This may result in high reflectivity or highdispersion with only a few grating pairs and makes possible the designand fabrication of compact narrowband reflectors, filters, switches, andmodulators for integration in photonic circuits. Such structures may beless sensitive to process variations in the formation of the gratingsacross a wafer than conventional gratings due to the multiple reflectionencounters with the same grating. Moreover, such DBR-MRR may not producethe reflection side-lobes that are prevalent in conventional DBRs. Theabsence of side-lobes may lead, for example, to reduced channelcrosstalk in dense wavelength division multiplexing (DWDM) systems andin lower power, higher-data-rate communications.

In an aspect, a device has an optical ring waveguide with a gratingalong at least a portion thereof, and an optical waveguide coupled tothe optical ring waveguide.

In another aspect, an optical source device includes an opticallyamplifying structure, where an optical ring waveguide may be coupled toat least one of an input or an output of the optically amplifyingstructure, the optical ring waveguide having a grating disposed along atleast a portion of the length thereof.

In yet another aspect, a laser having a gain medium may be adapted to bepumped by at least one of an optical or an electrical energy source;and, at least one optical ring waveguide having a grating along at leasta portion thereof. The optical ring waveguide is coupled to the gainmedium so as to form a resonant region including the gain medium.

An add-drop multiplexer is disclosed, including an optical ringwaveguide having a grating along at least a portion thereof a firstoptical waveguide coupled to the optical ring waveguide; and a secondoptical waveguide coupled to the optical ring waveguide. The add-dropwavelength is determined by a diameter of the optical ring waveguide anda period of the grating.

A method of designing an optical filter is disclosed, the methodincluding the steps of: selecting a design wavelength; determining acircumference of an optical ring waveguide to be an integral multiple ofthe design wavelength; selecting a length of a grating disposed along acircumference of the ring; selecting a reflection amplitude of a gratingdisposed along a length of the optical ring waveguide; and selecting acoupling coefficient between the ring optical waveguide and an opticalwaveguide coupled thereto; where the circumference, the couplingcoefficient and the reflection amplitude are selected to determine afilter spectral response.

A computer program product is disclosed for computing the parameters ofat least one of the reflective or transmissive optical filters, theproduct being stored in a non-volatile non-transient medium andincluding: instructions for configuring a computer to: accept dataincluding a design wavelength, a filter bandwidth and a filter maximumamplitude response; and, to compute a diameter of an optical ringwaveguide, a length of a grating disposed along a circumference of theoptical ring waveguide, an index contrast of said grating, a reflectioncoefficient spectra of said grating, and a coupling coefficient betweenthe ring waveguide and another optical waveguide such that that adesired filter bandwidth, and at least one of a reflection ortransmission coefficient, are achieved when the filter comprising theother waveguide coupled to the optical ring resonator is disposed in anoptical system.

The distributed reflector in a microring resonator architecture offers awide variety of functionalities and may lead to a significant reductionin photonic device size for equivalent optical performance. Aspects ofmicroring resonators and distributed Bragg reflectors may be combinedsuch that, for example, at resonance, the microring resonator enables ahigh intensity of optical field to build up in the ring. The grating mayproduce a specifically designed reflectivity spectrum that can enhanceor suppress reflection at the microring resonances. This enablesspecific reflectivity profiles for the entire structure such as, forexample, a single isolated peak or a periodic comb of peaks to beproduced.

A grating forming the reflector itself may be weakly reflecting (R<5%for a single pass through the grating at any wavelength), so that theoptical power in the optical ring waveguide at a design wavelength is atleast about 20 times larger than the power in the input bus opticalwaveguide. The weakly reflecting aspect of the grating near the designwavelength enables the DBR-MRR, for example, to effectively utilize thespectral filtering properties of a high-quality-factor microring.Otherwise, unwanted spectral mode splitting may occur and thereflectivity of the overall structure at the design wavelength issignificantly reduced

In devices such as the DBR-E(etalon)-MRR, described herein, the etalonstructure may have much higher reflectivity at a wavelength separatedfrom resonances of the microring (e.g. R>99% at a substantial wavelengthoffset, but R approaches 0% at a design wavelength, which is a resonanceof the microring).

Representative, but non-limiting attributes of a DBR-MRR component mayinclude:

-   -   a reflection spectrum similar to that of a distributed Bragg        reflector or sampled grating distributed Bragg reflectors        (SGDBR) with significantly smaller scalar dimensions (10-100×).        Reflection side lobes may be suppressed;    -   a periodic comb spectrum of reflection peaks with uniform or        non-uniform amplitudes which may be used, for example, as a        cavity-end-mirror in broad band quasi-continuous tunable lasers,        or other optical devices;    -   a single-wavelength reflection peak with high        side-mode-suppression-ratio (SMSR) and ultra-wide free spectral        range (FSR) which may be used, for example, as a        cavity-end-mirror for single-frequency lasers, or as a notch        filter for optical transmission.

A DBR-MRR may be configured to exhibit spectral amplitude and phaseresponses that may not be feasible using DBR, sampled grating DBR(SGDBR), or MRR structures either individually, or in combination, byincorporating, for example, DBR-etalon structures in the ring resonator(DBR-E-MRR). Such DBR-E-MRR structures may find applications in, forexample:

-   -   ultra narrow band transmission filters where the full width at        half maximum (FWHM) of the transmission pass-band may be, for        example, about 6 pm. This bandwidth is significantly narrower        than the 1040 pm FWHM for a MRR and 220 pm FWHM for an etalon,        when used individually. Such a narrow band filter can be used,        for example, to create a single frequency signal, to perform        optical analog signal processing, to modulate an optical signal        or to create a tunable narrow-linewidth filter for absorption        spectroscopy or other modes of sensing, or the like;    -   multiple output reflectors for compact semiconductor laser        cavities. A laser cavity with a two-optical-bus DBR-E-MRR mirror        as an output port may direct different portions of the laser        light and/or amplified spontaneous emission (ASE) to different        output ports. This may, for example, be used to reduce        spontaneous emission noise appearing at specific optical output        ports and enable configuration of a laser device exhibiting        lower relative intensity noise (RIN), which is useful, for        example, for low-power high-data-rate communications; and,    -   sharp transition mirrors, which may be used, for example, as        part of a passive external cavity for stabilizing or locking the        wavelength of a laser (the cavity would have dR/dλ<0 for        wavelength locking), or to modulate the location of a        transmission bandpass region, a reflection bandpass region or a        transition region in between, and use the device as an        amplitude, frequency, or phase modulated optical switch, or as        part of a tunable external cavity for modulating the wavelength        of a laser.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1A shows a representation of a DBR-MRR structure which may befabricated, for example, in on a substrate region that is 8 μm long by 9μm wide and achieves the same spectral response as B, a typical SGDBRmirror with 45 repetitions that occupies a 174 μm long region of width 5μm. The dimensions assume a 1 μm wide waveguide and 2 μm border per sidein both cases. Overall, the device area is 72 μm² for the DBR-MRR, whichis 12× smaller than the equivalent SGDBR of area 870 μm²;

FIG. 2 shows simulation results for the reflective structures describedin FIG. 1A, DBR-MRR, and B, SGDBR. Note that the DBR-MRR in A does notexhibit side-lobe ripples near the reflection peaks;

FIG. 3 shows the computed response of A, a half-DBR and B, a full-DBRring device where:

$z = {2{L_{t}\left( {\frac{n_{e}}{\lambda} - \frac{n_{e}}{\lambda_{0}}} \right)}}$is the normalized wavelength of light, L_(t) is the circumference of theDBR-MRR, n_(e) is the effective index of the ring waveguide, λ₀ is thedesign wavelength, and λ is the operating wavelength;

FIG. 4A shows an example where a single bus DBR-E-MRR produces a narrowtransmission window at a design wavelength: as shown in B nearresonance; and C, over a larger wavelength range, for m₁=1, m₂=1; anoff-resonance etalon for the single bus DBR-E-MRR, e.g. m₁=0.988 andm₂=1.012 produces a sharp transition mirror shown in D; and a double busDBR-E-MRR can produce a multiple-output reflector, where there may beoutput power at all four ports and a different spectral response at eachport, as shown in E;

FIG. 5A shows a scanning electron microscope (SEM) image of a partiallyprocessed half-ring DBR-MRR with apodized grating on the top half of thering; and B, a SEM image of a prototype full-ring DBR-MRR with grayscalegrating;

FIG. 6 shows a DBR-MRR schematic representation with a gap to the ringalong the circumference thereof;

FIG. 7 is a schematic representation of a quasi-continuously-tunablelaser with A, output along main laser waveguide, and B output alongseparate waveguide for filtering amplified spontaneous emission; C is aschematic representation of a narrow linewidth laser whereby the DBR-MRRselects a single wavelength for lasing: the second cavity reflector is abroadband high reflectivity coating on one end of the bus waveguide;and, D is a schematic representation of a quasi-continuously-tunablelaser where multiple DBR-MRR rings on each end of the lasing cavity arecascaded in series to achieve the desired comb profile, and to permitfine tuning as each ring can be tuned individually. Cascading of higherorder grating rings or etalon rings enable reflection spectra notavailable by a single ring;

FIG. 8 shows a schematic diagram of a microring resonator integratedwith a reflective element characterized by the scattering matrix S; thereflective element couples two modes propagating in opposite directions;superscripts + and − are used to denote the fields propagating in thedirection of the arrows and in the opposite direction, respectively;

FIGS. 9 A, B, C, and D show contour plots of the reflectance of theintegrated microring for different values of α² and τ² on the Θ-rdomain, respectively; A, α²=1, τ²=0.9; B, α²=1, τ²=0.8; C, α²=0.95,τ²=0.9; and D, α²=0.95, τ²=0.8;

FIG. 10 shows A, contour maps of critical reflection coefficient,r_(c)(0), and B, the resultant reflectance |a_(1c) ⁻|² from theintegrated microring on the τ²-α² plane;

FIG. 11 shows contour plot and reflectance spectra for a comb reflector:A, the plot of |a₁ ⁻(Θ,r)|² overlaid with the reflection profile of thereflective element r(Θ) (dashed line), with a periodic peak reflectionat Θ=2mπ; and B, reflectance spectra of the Fabry-Pérot MRR (FP-MRR) forvarious values of α². The Fabry-Perot (FP) reflection coefficients areset to their corresponding critical value r=r_(c)(0);

FIG. 12 shows A, a single peak reflector contour plot where thereflection profile of the reflective element (dashed line for half-ringDBR-MRR and dotted line for full-ring DBR-MRR) can be used to realize asingle peak reflector; and B, reflectance spectra of the single-peakDBR-MRR configuration;

FIG. 13 shows a schematic diagram of a DBR-E-MRR where each DBR mirrorelement is defined symmetrically by the dashed lines, which gives theextra length Λ for the ring portion;

FIG. 14 shows a narrowband transmission filter configuration obtainedusing the effective mirror model with A, the overlaid reflection profileof the etalon and B, the resultant transmission response. Under thelossless condition, the sum of reflection and transmission power at anypoint is unity;

FIG. 15 shows the response of a narrow transmission filter evaluatednumerically by a transfer matrix method (TMM) for A, the overlaidreflection profile; B, the resultant transmission response; C theamplitude transmission response as a function of λ; and D, thecorresponding phase transmission response;

FIG. 16 shows the response for a sharp transition mirror configuration;

FIG. 17 shows a cross section view of a DBR-MRR structure in the regionwhere the DBR-MRR structure is coupled to the bus waveguide;

FIG. 18 A shows a cross section of a structure suitable for quantum wellintermixing (QWI); and, B shows an example of coupling to a gain mediumusing a vertical coupling with a taper (with an inset view of theoverall structure); C, shows an enlarged perspective view of thestructure showing the DBR-MRR and the gain medium region; and, D showsan example of coupling from a gain region to the passive DBR-MRR locatedvertically above the gain region;

FIG. 19 shows an SEM image of an apodized GaAs grating: A, top viewshowing a full ring DBR-MRR, and, B angled side view showing an array of3 adjacent devices where the etched region appears white and thesidewalls of the ring and bus waveguide are visible;

FIG. 20 shows an example of a DBR-MRR using optical fiber, in aconfiguration where the DBR is separately fabricated and spliced into acircuit including a optical fiber directional coupler;

FIG. 21 shows the wavelength-dependent response computed for DBR-MRRconfigurations where the grating order is unity, and for higher ordergratings, for half-ring and full ring configurations;

FIG. 22 shows A, a schematic representation of a MRR with integratedDBR; B, a SEM image of a fabricated device prior to top claddingdeposition; C, a close up view of the portion of the image inside therectangle shown in B; and, D, an angled view of the modulated index ringand waveguide;

FIG. 23 shows the reflection and transmission spectra of a reflectivering resonator according to the model.

FIG. 24 shows A, the measured reflection and transmission spectra of areflective ring resonator; and B, an expanded view of the measuredspectra in A around the main reflection peak; the dashed lines show thesimulated spectra of a fitted model.

DESCRIPTION

Exemplary embodiments may be better understood with reference to thedrawings, but these examples are not intended to be of a limitingnature. Embodiments of this invention may be implemented in hardware,firmware, software, or any combination thereof, and may includeinstructions stored on a machine-readable medium, which may be read andexecuted by one or more processors.

When describing a particular example, the example may include aparticular feature, structure, or characteristic, but every example maynot necessarily include the particular feature, structure orcharacteristic. This should not be taken as a suggestion or implicationthat the features, structure or characteristics of two or more examplesshould not or could not be combined. When a particular feature,structure, or characteristic is described in connection with an example,a person skilled in the art may give effect to such feature, structureor characteristic in conjunction with other embodiments except when suchcombination is specifically excluded.

Reflective devices based on coupling of ring resonators have beenanalytically studied and experimentally demonstrated. At each resonancewavelength, an ideal microring resonator may support two degeneratecounter propagating resonance modes which are uncoupled. However, due tolarge field enhancement in a high Q ring, even a small perturbation cancreate strong coupling between the two modes. Mode coupling causessplitting of the two degenerate resonance wavelengths. Unwanted couplingcaused by imperfections in a resonator affects the performance ofmicroring based photonic devices and generally is consideredundesirable.

Herein, the spectral response properties of a grating in a ringresonator device are analyzed using two different computationaltechniques. In the first, the transfer matrix method (TMM) is used toderive a closed form one-dimensional analytical solution for fastsimulation in the steady state case. In the second, a graphicaltechnique enables an understanding of the effects of design parameterssuch as ring circumference, coupling factor, loss, DBR period, contrast,number of pairs, on the spectral response of a structure.

Using these techniques, a person of skill in the art having the benefitof this disclosure can design an optical device to have particularfunctionalities and advantages when compared to the individualcomponents thereof; for example: a single reflection peak with high SMSRand wide FSR, for example; a narrow full width at half maximumtransmission band, and a sharp spectral transition, with high extinctionratio and low insertion loss. Such devices may replace a SGDBR and mayhave additional useful properties. The one-dimensional computationalmodel has numerous simplifying assumptions and may often be used forinitial design purposes. To more accurately predict the responses and tomodel non-ideal effects such as refractive index dispersion, waveguidedispersion, scattering loss, and random fabrication imperfections,another technique based on coupled-mode theory may be used to obtain a3D approximation in cylindrical coordinates. The calculation of thecoupling of counter propagating fields due to arbitrary perturbationsalong the ring shows results comparable to finite element method (FEM)results but generally requires considerably less time to process.

The use of the graphical design method and the coupled-mode approach isdescribed herein along with fabrication techniques and simulationresults. Other design approaches can also be used, such as FEM.

An optical mirror device comprising a single microring resonator with anintegrated reflective element is described along with a graphical methodto engineer overall frequency-dependent response without needingdetailed numerical computations, at least for preliminary design. Thisanalysis of a microring resonator with a distributed Bragg reflectorintegrated into or coupled thereto may be applied to more generalreflective elements as well, including chirped gratings and gratingetalon combinations. Configurations of a comb reflector, a single-peakreflector, a narrow transmission filter, and a sharp transition mirrorare described as examples. Such devices may have typical dimensions of amicroring resonator, allowing dense integration of the elements asplanar photonic devices.

In FIG. 1A, a MRR 10 is coupled to another optical waveguide 20 to forma distributed Bragg reflector in a microring resonator 5. This couplingis shown schematically as having a portion of the MRR 10 being inproximity to a region of the optical waveguide 20. A person of skill inthe art would appreciate that the coupling coefficient κ² between theMRR 10 and the optical waveguide 20 would depend, for example, on theproximity of the two structures in wavelengths, the refractive indiciesof the two structures, and that of any cladding or substrate medium onwhich the other structures are formed.

FIG. 1B shows a typical SGDBR 30, comprising a plurality of DBR regions50 spaced along its length. In this example, each of the DBR regions isidentical, although other configurations are known and may be used. Eachof the DBR regions 50 may include a plurality of modified refractiveindex elements having a refractive index n₂, spaced apart along thelength of the DBR region, where the native refractive index of thewaveguide is n₁. In this configuration, there may be a region ofrefractive index n₁ separating the periodic bursts of refractive indexn₂. The distance between the start of successive DBR regions isZ_(SGDBR), and the overall length of the SGDBR is L_(SGDBR).

The reflectivity spectrum of a DBR-MRR 5 can be engineered to be similarto that of a much longer SGDBR 30. Using the DBR structure of FIG. 1Awhere the circumference Z of the microring resonator 10 is twice theperiod Z of the SGDBR (Z_(MRR)=²Z_(SGDBR)), an equivalent FSR may beachieved. For a given L_(SGDBR), the FWHM of the two structures may bematched by choosing an appropriate coupling coefficient κ² between theMRR 10 and the optical waveguide 20, which affects the amount of fieldbuilt up in the ring; that is, a quality factor, Q, of the ring. TheFWHM may be reduced without increasing device size, an attribute whichmay not be feasible in a SGDBR. The peak reflectivity of each peak in aDBR-MRR can be controlled by adjusting, for example, the index contrastn₂-n₁ and other parameters of the grating in the ring.

Also, the multiple reflection encounters between the optical field andeach of the refractive index perturbations of the grating in the MRRring 10 may relax the requirement for tight fabrication tolerancesordinarily requiring uniformity of grating dimensions across a wafer.Moreover, as will be seen, the reflection spectral side lobes may beeffectively suppressed in embodiments of the DBR-MRR. Generally, theoutput facets of the coupled waveguide 20 should be coated with ananti-reflective coating or angled facet so as to suppress unwantedreflections therefrom at a waveguide air interface, or otherdiscontinuity. However, in circumstances such as where a resonant cavityis to be formed, one or more of the facets may be configured to providefor a reflection in lieu of using a MRR. Such an example would be alaser device as described herein.

A single peak in reflection is usually achieved by a long, continuousDBR, or SGDBR. The equivalent DBR-MRR structure is much smaller indimensions, may also achieve a narrower FWHM, effectively eliminatereflection side lobes, and is less sensitive to fabrication tolerances.Moreover, the filter roll-off may be faster than conventional lineargratings as illustrated in FIG. 24.

In an example, a very long DBR structure with continuous gratings (116μm long by 5 μm wide, not shown here) could achieve a single reflectionpeak. Such a device structure may be replaced by an equivalent, but muchsmaller half-ring DBR-MRR or a full-ring DBR-MRR (8 μm long by 9 μmwide) resulting in an 8× reduction in substrate area. FIG. 2A shows thespectral response of the DBR-MRR and FIG. 2B shows the spectral responseof the equivalent SGDBR. The term “ring” is used with the understandingthat the “ring” need not be circular. A racetrack shape, or othertopologically equivalent shape, is usable providing only that the radiusof curvature along the ring is sufficiently large with respect to theoperating wavelength so as to keep radiative losses within design limitsfor overall loss. Where the terms “half-ring” and “full-ring” are used,they are meant to figuratively describe the situations shown in FIGS. 3Aand 3B, respectively, where the overall linear extent of the DBR iscompared with the circumference of the MRR.

Using the half-ring DBR-MRR and the full ring DBR-MRR configurations asshown in FIGS. 3A and 3B, respectively, the side-lobe ripple of theDBR-MRR configuration near z=0 may be effectively suppressed. Althoughideal half-ring and full-ring DBR-MRR devices have a perfect null inreflectivity at non-zero even integers of z, the ring resonance mayresult in weak reflections near these values.

By inserting an etalon structure in the MRR 10 with a gap equal to ahalf-integer multiple of the effective design wavelength λ₀ (FIG. 4A), anarrow transmission filter may be obtained. When the length of theetalon in the single bus configuration is chosen to be offset from theresonance of the ring, the output characteristics are such that sharptransition mirror is obtained.

A single bus DBR-E-MRR forming a narrow transmission window at thedesign wavelength as shown in FIG. 4C for m₁=1, m₂=1. The response farfrom the design wavelength is shown in FIG. 4D. An off-resonance etalonfor the single bus DBR-E-MRR, e.g. m₁=0.988 and m₂=1.012 would yield asharp transition mirror shown in FIG. 4E. A configuration adding anotherbus waveguide 50 coupled to the DBR-E-MRR as shown in FIG. 4E to form adouble bus device, can be used to produce a multiple-output reflector,where there may be output power at all four ports and a differentspectral response at each port, and whose response is shown in FIG. 4F.This device may, for example, be used as an add-drop multiplexer inoptical networks.

The gratings on the microring resonators may be fabricated, for example,either vertically (two-step or grayscale lithography) or horizontally(apodized). Other fabrication techniques may be used as well, as are nowknown or may be subsequently be developed. Here, apodization refers tovarying the width of the waveguide. This apodization of width may berectangular, sinusoidal, triangular, or another shape so as to achieve adesired reflection strength and spectral profile. Vertical gratings maybe fabricated using a two-step known method using vertical anisotropicetching, resulting in difference in height of the material. The two-stepapproach begins with patterning and etching the ring 10 and thewaveguide 20 in a substrate. Next, the substrate surface is planarizedwith a material such as benzocyclobutene (BCB) polymeric resin. Thegratings are then patterned on the ring 10, and a second etch performed.

The grayscale method involves patterning the ring 10 and waveguide 20using a high exposure dose and the gratings using a low exposure dose,in a single manufacturing stage. This process creates two different maskheights, one for the waveguide and ring, and another for the gratings.By transferring the different mask heights to the wafer during etching,one may achieve vertical gratings on a microring without a planarizationprocess or a second etching step.

Apodized gratings may be generally easier to fabricate than the verticalgratings. When patterning the waveguide and the ring, the gratings maybe incorporated in the pattern, and with a single patterning and etchingstep, the gratings are fabricated along with the waveguide and the ring(see FIGS. 5A, B). With apodized gratings, the refractive index contrastof the gratings may be manipulated because the index contrast iscontrolled primarily by the pattern rather than a fabrication parametersuch as etch depth. Other apodizing techniques may be used.

Bragg gratings may be directly patterned on the microring resonator(e.g., by apodized or vertical etch) or separated by a small distancefrom the ring along its circumference as shown in FIG. 6. In eachgeometry, the strength of the reflection coefficient from the gratingitself can be controlled. In the apodized example, this may be done bysetting the change in width of the apodization (Δw). In the verticaletch embodiment, the etch depth can be controlled. In the separatedgrating embodiment, the gap between the ring and the grating teeth (g₁)as well as the width of the grating teeth (W_(c)) may be used to similareffect. In the apodized and in the separated embodiments, the gratingsmay be patterned on the inner side of the ring, the outer side of thering, or both. This provides an additional technique which may be usedto control the reflection coefficient.

The period of the grating around the circumference may be a constantvalue so that the grating provides strong reflection at a singlewavelength, or it may be chirped or sampled so that the grating providesstrong reflection or phase shifts at multiple wavelengths. The combinedresponse of the ring and grating may be significantly different thanthat of the grating alone due to the resonant nature of the ring. Whiledetails of these gratings are not described herein, they will befamiliar to a person of skill in the art, who may use such grating inaccordance with the teachings herein to produce a variety of opticalcomponents and devices.

DBR-MRR optical elements may be used, for example, as cavity mirrors forcompact narrow-linewidth single-frequency or tunable laser sources. Asshown in FIG. 7A, a compact tunable laser 310 can be comprised ofoptical elements having parameters selected to control the gain 370 and,the phase 385 of the cavity and also the center wavelengths of each ofthe DBR-MRR end mirrors 365. The end mirror may be, for example, asingle λ/4 DBR-MRR comb mirror 365, so that a comb reflectivity profileis generated to enable quasi-continuous tuning using the Vernier effect.The grating of the DBR-MRR may be a higher order grating such as 3λ/4,5λ/4, . . . , in order to select a desired subset of peaks from thecomb. That is, the DBR-MRR 365 may be designed to be reflective at allresonances of the microring, at every other resonance, at every thirdresonance, or as desired. A comb version of the DBR-E-MRR (long etalons)can be used to produce a multiple-output-port mirror, as shown in FIG.7B. The phase-control section can be a MRR 385, or other opticalelement, or a portion of the coupled waveguide 20 for electrooptical orthermal tuning 410 (as shown in FIG. 7C) while providing a low insertionloss (see, for example, “Micrometer-scale silicon electro-opticmodulator”, by Qianfan Xu, Nature Letters, Vol. 435, 19 May 2005 whichis incorporated herein by reference). A doped-waveguide gain section 370is disposed between the mirrors 365 and is pumped by well knowntechniques (not shown). Low-threshold lasers are possible since themirror reflectivities can be high and thus the gain cavity 370 can besmall; the device footprint may be 100× smaller than conventional DBRtunable lasers. One of the end mirrors may be a cleaved or etched facetwith a highly reflective coating (see FIG. 7C, 430).

One end mirror may be a DBR-MRR structure (e.g., 365), and the other endmirror is either a cleaved facet, etched facet, or a cleaved or etchedfacet with a broadband highly reflective (HR) coating (see FIG. 7C,430). The broadband HR coating may be formed, for example, by depositionof a SiNx and SiO2 dielectric Bragg stack on the facet. This alternategeometry, which may be used in any of the examples, obviates the needingto align the DBR-MRR wavelengths on the two cavity ends to be the same,and may simplify fabrication. Such a configuration can be used, forexample, to make single wavelength lasers, tunable lasers, or laserswith other desirable features, including active optical sensors or otheractive photonic integrated circuit elements.

As shown in FIG. 7D, the grating may occupy some fraction of the ring,rather than the half ring or the full ring. This arrangement may resultin a reflection peak of unequal amplitude at each resonance. Asdescribed below, a higher-order grating may be used to generate unequalamplitude peaks at every q^(th) resonance, where q=1, 2, 3, is somechosen natural number. FIG. 7D also illustrates that, by cascadingseveral micro rings with different diameters and different centerwavelengths (1 and 1′ are close in frequency, 2 and 2′ are close infrequency, 3 and 3′ are close in frequency), the exact comb pattern andtuning can be better controlled better.

While the examples herein often use waveguide elements fabricated on anoptical substrate, optical waveguides may also include optical fiber,which may be doped to provide a gain section or a phase control section,as well as the DBR-MRR, where the grating may be written by opticalbleaching or other techniques. Composite designs using componentsfabricated on an optical substrate, optical fiber, semiconductor devicesand the like may also be used to achieve the characteristics describedherein.

In an example, a method of design of a single microring resonator 10with an integrated reflective element S is described, using thegraphical method. The ring 10 has a total circumference L_(t)=L+L_(s),where L and L_(s) correspond to the length of the ring 10 and thereflective element S respectively, as depicted in FIG. 8. Allelectromagnetic fields are assumed to be normalized in power.

For simplicity, the coupling and transmission coefficients, κ and τ, ofthe ring-bus coupler and the round trip field attenuation in the ring αare assumed to be wavelength independent, although this is not intendedto be a limitation. The analysis presented may be extended to a moreaccurate performance model which may involve wavelength dependentcoefficients, so as to account for refractive index and waveguidedispersions.

In an example, consider a passive case, 0<α≦1. While this is a passiveexample, this is not intended to preclude the incorporation of gainelements such as semiconductor optical amplifiers, doped-fiber opticalamplifiers, and the like into the structure, such as has been previouslydescribed. Where the gain element is a semiconductor device, the devicemay be treated as a lumped device and incorporated in the scatteringmatrix. Doped-fiber components may use, for example Erbium, Neodymium orother rare earth elements in combinations that are known or may bedeveloped for producing optical gain over a bandwidth when pumped by asource of energy. For the purposes of this discussion, thecharacteristics of the gain element may be lumped with the scatteringmatrix S.

The scattering matrix of the integrated reflective element be expressedas given

$\begin{matrix}{S = \begin{pmatrix}S_{11} & S_{12} \\S_{21} & S_{22}\end{pmatrix}} & (1)\end{matrix}$

If the reflective element is assumed to be lossless, passive andreciprocal, the scattering matrix is:

$\begin{matrix}{S = {\begin{pmatrix}{{\mathbb{i}}\; r\;{\mathbb{e}}^{\mathbb{i}\psi}} & t \\t & {{\mathbb{i}}\; r\;{\mathbb{e}}^{- {\mathbb{i}\psi}}}\end{pmatrix}{\mathbb{e}}^{\mathbb{i}\phi}}} & (2)\end{matrix}$which satisfies the lossless condition S^(†)S=I and the reciprocitycondition S₁₂=S₂₁. We use the e^(−iωT) time convention. Here, r and tare the magnitude of the reflection and transmission coefficients of theintegrated reflective element, respectively, such that r²+t²=1, and φ, ψare phase terms.

The steady state solutions of the transmitted and reflected fieldsnormalized to the input signal of the microring are:

$\begin{matrix}{b_{1}^{+} = \frac{\tau - {\alpha\;{t\left( {1 + \tau^{2}} \right)}{\mathbb{e}}^{{\mathbb{i}}{({\theta + \phi})}}} + {\alpha^{2}{\tau\mathbb{e}}^{{\mathbb{i}2}{({\theta + \phi})}}}}{1 - {2\alpha\; t\;{\tau\mathbb{e}}^{{\mathbb{i}}{({\theta + \phi})}}} + {\alpha^{2}\tau^{2}{\mathbb{e}}^{{\mathbb{i}2}{({\theta + \phi})}}}}} & (3) \\{{a_{1}^{-} = \frac{{\mathbb{i}\alpha}\;{r\left( {1 - \tau^{2}} \right)}{\mathbb{e}}^{{\mathbb{i}}{({\theta + \phi - \psi})}}}{1 - {2\alpha\; t\;{\tau\mathbb{e}}^{{\mathbb{i}}{({\theta + \phi})}}} + {\alpha^{2}\tau^{2}{\mathbb{e}}^{{\mathbb{i}2}{({\theta + \phi})}}}}}{{{where}\mspace{14mu} K} = \begin{pmatrix}\tau & {\mathbb{i}\kappa} \\{\mathbb{i}\kappa} & \tau\end{pmatrix}}} & (4)\end{matrix}$

is the coupling matrix; κ, τ are amplitudes of coupling and transmissioncoefficients, respectively; a₁ ⁺=1, b₁ ⁻=0; and, a lossless couplingcondition, κ²+τ²=1, are assumed where:φ=∠S ₁₂  (5)θ=βL  (6)are the transmission phase shifts of the reflective element and the ringportion, respectively, and β is the modal propagation constant in thering waveguide.Θ=(θ−θ₀)+(φ−φ₀)  (7)is defined as total detuned round trip phase shift in the integratedmicro-ring, where the subscript represents the quantity at the designwavelength λ₀. Unless otherwise specified, L is chosen to satisfyθ₀+φ₀=2mπ for some integer m (resonance at the design wavelength), andthus a₁(θ+φ)=a₁ ⁻(Θ) and b₁ ⁺(θ+φ)=b₁ ⁺(Θ).

Although Eqs. (3) and (4) were solved assuming the reflective element ispositioned at the center of the ring, the reflection and transmissionfield amplitude |a₁ ⁻| and |b₁ ⁺| in the model are not dependent on therelative position of the integrated element in the ring, provided thatthe inserted element is not directly coupled to the bus waveguide. Onlythe phase of the reflection coefficient ∠a₁ ⁻ is affected by therelative position of the reflective element. The transmission phase ∠b₁⁺ is not affected.

If the attenuation coefficient cc of the microring and the transmissioncoefficient τ of the ring-bus waveguide coupler are given parameters ofthe microring, r(Θ) and t(Θ) may be engineered using Eqs. (3) and (4). ψmay be neglected, as it does not affect the output power as seen fromEqs. (4). For a lossless reflective element, t=√{square root over(1−r²)} and therefore Eqs. (3) and (4) need evaluation only in the Θ-rdomain. The condition for obtaining the maximum reflection that can berealized from the integrated microring for given α and τ may be found bysolving the differential equation ∇_(Θ,r)|a₁ ⁻(Θ,r)|₂=0, which yieldsthe reflective element critical reflection coefficient profile

$\begin{matrix}{{r_{c}(\theta)} = \frac{\sqrt{\left( {1 - {\alpha^{2}\tau^{2}}} \right)^{2} + {4\alpha^{2}\tau^{2}\sin^{2}\Theta}}}{1 + {\alpha^{2}\tau^{2}}}} & (8)\end{matrix}$for −π/2≦Θ+2mπ≦π/2. The resultant maximum reflection amplitude may beevaluated by substituting Eq. (8) into Eq. (4):

$\begin{matrix}{{a_{1c}^{-}} = \frac{\alpha\left( {1 - \tau^{2}} \right)}{1 - {\alpha^{2}\tau^{2}}}} & (9)\end{matrix}$

For constant values of α and τ, a maximum reflectance |a_(1c) ⁻|² existsand is the same for continuous points in the Θ-r plane. At resonanceΘ=2mπ, Eq. (8) reduces to

$\begin{matrix}{{r_{c}(0)} = \frac{1 - {\alpha^{2}\tau^{2}}}{1 + {\alpha^{2}\tau^{2}}}} & (10)\end{matrix}$which is the minimum field reflection amplitude necessary to produce amaximum reflectance from the microring. For α²≈1 and κ²<<1, Eq. (10)reduces to r_(c)(0)≈κ²/2.

Note that the reflectance |a₁ ⁻|² is proportional to the field buildupintensity of the − mode in the ring. A large value of |a₁ ⁻|² isevidence that the mode is in resonance in the structure. Therefore, thebranching off of the peak reflection condition r_(c)(Θ) in FIG. 9 forr_(c)(Θ)>r_(c)(0) corresponds to the resonance-splitting and theseparation between resonant peaks can be obtained by calculating thecorresponding 0 from Eq. (8) provided that α, r, and τ are known.

The reflectance amplification factor may be written as:

$\begin{matrix}{A = {\frac{{a_{1c}^{-}}^{2}}{r_{c}^{2}(0)} = \frac{{\alpha^{2}\left( {1 - \tau^{2}} \right)}^{2}\left( {1 + {\alpha^{2}\tau^{2}}} \right)^{2}}{\left( {1 - {\alpha^{2}\tau^{2}}} \right)^{4}}}} & (11)\end{matrix}$which represents the maximum enhancement in the reflected power due tothe field buildup in the ring. The factor A can be quite large for lowloss devices and thus can be used for high sensitivity sensorapplications.

As α or τ decreases, the full width half maximum of each individual peakincreases (see FIG. 9).

FIG. 10A depicts contour lines of the minimum critical reflectionr_(c)(0) and the resultant reflectance from the ring |a_(1c) ⁻|² on theτ²-α² plane. For example, if τ²=0.9 and α²=1, we obtain r_(c)(0)=0.053and |a_(1c) ⁻|²=1 from FIG. 10, which agrees with the values from FIG.9A. Note that r_(c)(0) increases as α or τ decreases, and |a_(1c) ⁻|²decreases as a decreases or τ increases. Thus, increasing α increasesthe amplification factor A whereas increasing τ can either increase ordecrease A.

Further, for a given integrated reflective element, r and Θ are relatedto each other. One can construct r(Θ), the reflection profile of theintegrated element, as a parametric function of the total detunedround-trip-phase from |a₁ ⁻(Θ,r)| φ(λ), and θ(λ). By overlaying the r(Θ)profile on the contour plot of |a₁ ⁻(Θ,r)| on the same Θ-r plane, theeffect of the reflective element on the reflectance of the integratedmicroring resonator may be visualized.

In additional examples, configurations of wavelength-selective devicesobtained by designing r(Θ) of the reflective element using the graphicalsolutions are described. These examples use an integrated microringresonator with the parameters L_(t)=100λ_(e,0) and τ²=0.9, whereλ_(e,0)=λ₀/n_(e) is the effective design wavelength in the waveguide.The selection of the parameters is for specificity in describing theexamples, so as to improve understanding of the details, and is notmeant as a limitation.

An ideal comb reflector has high reflectance peaks spaced periodically.From the periodicity of the |a₁ ⁻|² contour plot, a comb reflector maybe obtained by using a spectrally flat r(Θ). FIG. 11A shows this case.The dashed line represents the desired reflection profile of thereflective element r(Θ) overlaid on the contour plot, with a constantvalue of r_(c)(0)=0.053 for α²=1. The result is a periodic reflectionspectrum from the microring with peaks at Θ=2mπ.

One of the physical candidates for the reflective element to achievesuch a flat reflection profile is a low-reflectivity FP element. For alow-reflectivity FP element, the phase response is approximately alinear function of β with the slope equal to its length d; by choosingL>>d, the reflection may be a constant value.

Consider a simple index-contrast FP element. The reflection from anindex-constant FP element with length L_(s)=d=¼(λ_(e,0)). is given by

$\begin{matrix}{{S_{11}}^{2} = \frac{4R_{int}\sin^{2}\beta\; d}{\left( {1 - R_{int}} \right)^{2} + {4R_{int}\sin^{2}\beta\; d}}} & (12)\end{matrix}$where R_(int)=r_(int) ² is the reflection power from each interface. Atthe design wavelength, Eq. (12) can be expressed as:

$\begin{matrix}{r = \frac{2\sqrt{R_{int}}}{1 + R_{int}}} & (13)\end{matrix}$

The transmission coefficient of the FP element is given by:

$\begin{matrix}{S_{12} = \frac{\left( {1 - R_{int}} \right){\mathbb{e}}^{{\mathbb{i}\beta}\; d}}{1 - {R_{int}{\mathbb{e}}^{{\mathbb{i}}\; 2\beta\; d}}}} & (14)\end{matrix}$

From Eqs. (5) and (14), one may observe that φ(β)≈βd for R_(int)<<1,which yields the partial derivative

$\frac{\partial\phi}{\partial\theta}{\operatorname{<<}1}\mspace{14mu}{since}\mspace{14mu} d{\operatorname{<<}{L.}}$The reflection profile of the FP element may be modeled as a constant,r(Θ)≈r_(c)(0), near the design wavelength. Since the phase response inthe ring is approximately linear, one may choose L_(t) to be an integernumber of λ_(e,0), which yields θ₀+φ₀=200π, in this example.

FIG. 11B shows the reflectance spectra of the FP-MRR for α²=1, 0.99, and0.95. For each value of α², the FP reflection is set tor=r_(c)(0)=0.053, 0.058, and 0.078 respectively. The peak reflection ofeach FP-MRR corresponds to |a_(1c) ⁻|²=1, 0.83, and 0.45, respectively.Thus, the ring resonator amplifies the reflectance by A=360, 250, and74, respectively.

For lossy cases, a decrease in τ results in an increase in the criticalreflection coefficient r_(c)(0) and the peak reflectance |a_(1c) ⁻|².However, the increase in r_(c)(0) may require a higher value of R_(int)from Eq. (13). A higher value of R_(int) may cause deviation from thelinear phase approximation of the FP element. The FWHM values of thereflection intensity of the FP-MRRs are obtained to be (2.4×10⁻⁴)λ₀,(2.6×10⁻⁴)λ₀, and (3.2×10⁻⁴)λ₀ for α²=1, 0.99, and 0.95, respectively atthe design wavelength. In comparison, the FWHM values of buildup fieldintensity of the identical microrings with no internal reflection, i.e.,r(Θ)=0, are calculated to be (1.7×10⁻⁴)λ₀, (1.8×10⁻⁴)λ₀, and(2.5×10⁻⁴)λ₀ for the same set of α² values.

The reflectance spectra of an FP-MRR resembles that of a SGDBR, but theFP-MRR may not exhibit side lobes. Furthermore, the FWHM may bedetermined by the micro-ring parameters α² and τ², allowing anadditional degree of freedom in engineering the peak shape. Some of theperformance differences with respect the SGDBR may be more compactdimensions, simpler architecture, faster filter roll-off and a reducedsensitivity to wafer scale variations due to reuse of the same FPreflector element.

A single-peak reflector element at the design wavelength may be designedby suppressing the reflection r(Θ) of the reflective element at thering-resonant-wavelengths other than the design wavelength. That is, thereflective element whose reflection profile satisfies

$\begin{matrix}{{r(\Theta)} = \left\{ \begin{matrix}0 & {{{{for}\mspace{14mu}\Theta} = {2m\;\pi}},{m \neq 0}} \\{r_{c}(0)} & {{{for}\mspace{14mu}\Theta} = 0.}\end{matrix} \right.} & (15)\end{matrix}$

In an index-contrast N-period Bragg grating as a reflective element, thetotal length of the grating is L_(s)=L_(g)=NΛ≈Nλ_(e,0)/2, where Λ is theperiod of the grating. In the low reflectivity limit, N√{square rootover (R_(int))}<0.2 and, neglecting multiple reflections in the grating,one may obtain the reflection coefficient

$\begin{matrix}{{S_{11}} \approx {2N\sqrt{R_{int}}{\frac{\sin\;\Theta_{g}}{\Theta_{g}}}}} & (16)\end{matrix}$where Θ_(g)=(β−β₀)L_(g) is the detuned phase shift in the grating. Thetransmission phase shift of the low-reflectivity grating isapproximately a linear function φ(β)≈βL_(g), from which one may obtainΘ≈(β−β₀)L_(t). To satisfy Eq. (15), the grating contrast and thefraction of the ring occupied by the grating may be controlled.p=Θ _(g)/Θ  (17)is the ratio between the detuned phases of the grating to the entiremicroring. Then,

$\begin{matrix}{0 < p \approx \frac{L_{g}}{L + L_{g}} \leq 1.} & (18)\end{matrix}$Note that p is fixed for a given geometry of the structure. Eq. (16) maybe written as

$\begin{matrix}{{r(\Theta)} = {2N\sqrt{R_{int}}{\frac{\sin\left( {p\;\Theta} \right)}{p\;\Theta}}}} & (19)\end{matrix}$And, imposing the first condition in Eq. (15) on Eq. (19) yieldssin(pΘ)=0 for all Θ=2mπ, m≠0, so 2 mpπ should be an integer multiple ofπ. Therefore, from Eq. (18) possible choices for p in this example arep=½ or p=1. The corresponding grating lengths may be

$\begin{matrix}{{L_{g} = {L = {{\frac{1}{2}L_{t}\mspace{14mu}{or}\mspace{14mu} L_{g}} = L_{t}}}},{L = 0}} & (20)\end{matrix}$Physically, the first case is where the grating occupies nominally halfof the ring (FIG. 3A), and the second case is where the gratingnominally occupies the entire ring (FIG. 3B). Exact values of p for asingle wavelength reflector can be recalculated numerically if theassumptions above are not satisfied. Other values of p may be used indifferent applications.

To meet the second condition in Eq. (15), the appropriate gratingcontrast to set R_(int) and satisfy 2N√{square root over(R_(int))}=r_(c)(0) is chosen. The reflection profile for each case isoverlaid on the contour plot in FIG. 12A. The full grating has nulls,i.e., r=0 at Θ=mπ for all nonzero integers m where the half-ring hasnulls at Θ=2 mπ. FIG. 12( b) depicts the resultant reflectance spectraof the two DBR-MRRs for α²=1. It should be noted, however, that for thefull DBR-MRR, the grating at the coupling region may cause somereflection and scattering.

There is competition between the field buildup due to resonance anddecaying reflection from the grating near Θ=2mπ, m≠0. However, theoverall microring reflection will approach zero as the gratingreflection coefficient goes to zero, thus suppressing reflection atadjacent ring resonances. As a result, only a single peak at the designwavelength is observed.

The FWHM of the reflection intensity is (2.4×10⁻⁴)λ₀ for both the halfDBR-MRR and the full DBR-MRR at the design wavelength, for the losslesscase. Compared to a continuous DBR, the DBR-MRR may possess advantagessuch as a more compact structure, suppression of side-mode ripples, theability to design the FWHM from the microring parameters, faster filterroll-off and reduced sensitivity to wafer-scale variations.

If the interface reflectivities of the FP element are increased, it ispossible to construct a narrow band-pass filter by employing a sharpphase change of the FP element near the resonance condition thereof.However, a highly reflective FP element with a single material interfacemay be somewhat difficult to fabricate; instead, one can employ highreflection DBRs to form a DBR etalon. A microring resonator integratedwith a high reflection DBR etalon may be termed a DBR-E-MRR. In thisregime, the phase response of the reflective element is no longerlinear; that is, the approximation φ(β)≈βL_(s) no longer holds.

Consider an integrated microring structure which includes two identicalN-period DBRs on the left and the right half of the ring and gaps oflength d and L. Assume that d is an integer multiple of the gratingperiod Λ. To make the analysis simpler, we re-define the DBR structureto be symmetric; that is, the last half-period portion at the bottom isexcluded, as shown in FIG. 13.

The length of the new DBR is then

$\overset{\_}{L_{g}} = {\left( {N - \frac{1}{2}} \right){\Lambda.}}$By doing so, an additional condition is imposed on S₁₁=S₂₂ in Eq. (2)and the grating scattering matrix is obtained.

$\begin{matrix}{{S_{g}\begin{pmatrix}{\pm r_{g}} & {it}_{g} \\{it}_{g} & {\pm r_{g}}\end{pmatrix}}{\mathbb{e}}^{{\mathbb{i}\phi}_{g}}} & (21)\end{matrix}$where r_(g) and t_(g) are the magnitude of reflection and transmissioncoefficients of the grating, and φ_(g) corresponds to the phase of thereflection coefficient. The + or − sign is taken when the last index ofthe DBR is high or low, respectively. Note that with the symmetric DBRdefinition, the length of the ring portion was L=L+Λ as shown in FIG.13. We define θ=β L and the transmission coefficient of the upperDBR-etalon structure of length d+2 L_(g) becomes

$\begin{matrix}{S_{12} = \frac{{- t_{g}^{2}}{\mathbb{e}}^{{\mathbb{i}2\phi}\; g}{\mathbb{e}}^{{\mathbb{i}\beta}\; d}}{1 - {r_{g}^{2}{\mathbb{e}}^{{\mathbb{i}2\phi}\; g}{\mathbb{e}}^{{\mathbb{i}2\beta}\; d}}}} & (22) \\{{{\angle\; S_{12}} = {{{atan}\left\lbrack {{\Gamma tan}\left( {{\beta\; d} + {\phi\; g}} \right)} \right\rbrack} + \phi_{g} + \pi}}{{{where}\mspace{14mu}\Gamma} = {{{\frac{1 + r_{g}^{2}}{1 - r_{g}^{2}}\mspace{14mu}{and}\mspace{14mu} r_{g}^{2}} + t_{g}^{2}} = 1.}}} & (23)\end{matrix}$From Eq. (23), we obtain

${{\beta\; d} + \phi_{g}} = {{{{atan}\left\lbrack {\frac{1}{\Gamma}{\tan\left( {\phi - \phi_{g}} \right)}} \right\rbrack}\mspace{14mu}{where}\mspace{14mu}\phi} = {\angle\;{S_{12}.}}}$

The reflection power from the DBR-etalon is given by:

$\begin{matrix}{{r\left( {\phi - \phi_{g}} \right)} = {\frac{2\sqrt{R_{g}}}{1 + R_{g}}{{\sin\left( {\phi - \phi_{g}} \right)}}}} & (24)\end{matrix}$which is the reflection profile of an integrated DBR-etalon structure.

The reflection profile is a function of the total detuned phase shift inthe integrated ring Θ=(θ−θ₀)+(φ−φ₀) requires further analysis as thegrating phase response φ_(g) is not linear. One may employ a numericalsimulation such as the transfer matrix method to obtain the reflectionprofile r(Θ) of the DBR-etalon as a function of the total detuned phase.

FIG. 14 shows the response of a DBR-E-MRR under lossless conditions withr_(g) ²=0.99, d=L=0, and N=100 using a linear approximation.

FIG. 15 shows the transmission response of the device using the TMM. TheFWHM of the isolated DBR etalon δΘ_(DBR-etalon)≈0.34π is reduced toδΘ_(DBR-E-R)≈0.016π by integrating the DBR etalon combination into amicroring. To find the microring response as a function of λ, substituteΘ(β)=[(d+2L_(e))Γ+2L_(e)+L+Λ)](β−β₀), and note that β/β₀=λ₀/λ, assumingdispersion is negligible, for simplicity. The slope∂Θ/∂β=4L_(e)/(1−R_(g))+Λ can be large when the effective grating lengthL_(e) is long or the reflection R_(g) is large, and this yields a sharpresponse as a function of β or δΘ. For example, δΘ=0.016π corresponds

${\delta\lambda} = {{{\frac{\partial\lambda}{\partial\beta}\frac{\partial\beta}{\partial\Theta}{\delta\Theta}}} \approx {\left( {\frac{1 - R_{g}}{4\beta_{0}L_{e}}{\delta\Theta}} \right)\lambda_{0}} \approx {\left( {2.4 \times 10^{- 6}} \right)\lambda_{0}}}$to in this particular case. The FWHM in wavelength decreases rapidly asR_(g)→>1. The value of R_(g)=0.99 was chosen considering typical maximumreflectance values of planar DBRs.

When d=0, we get r=0 from Eq. (24) if φ_(g)=mπ for any integer m. Thiscorresponds to Θ=(β−β₀)Λ+2mπ since φ_(g)(β₀)=0. That is, for

${{{\left( {\beta - \beta_{0}} \right)\Lambda}}{\operatorname{<<}{\pi ❘}}},{i.e.},{{{\frac{\beta}{\beta_{0}} - 1}}{\operatorname{<<}1}},$the first term in Θ becomes negligible, and therefore the DBR-etalon hasa null at the adjacent ring resonances in addition to the null at λ₀.

If L is detuned to L′=L+ΔL, we obtain θ′=θ+θ+(β−β₀)ΔL, which in turnshifts Θ′=Θ+(β−β₀)ΔL; that is, increasing L shifts r(Θ) horizontally tothe right and increases the slope ∂Θ′/∂β=∂Θ/∂β+ΔL by a negligibleamount. FIG. 16 depicts the spectral response of the micro-ring deviceunder the lossless condition, r_(g) ²=0.5, and ΔL=0.011λ_(e,0). Thesevalues were chosen for the example so that the etalon reflectioncoefficient curve (dashed) is approximately tangent to the criticalreflection curve r_(c). This produces a spectrally widerhigh-reflectance band. Adjustments to the parameters can be made toreduce the ripple at the expense of a narrower high-reflectance band.Note that the null of the etalon reflection profile corresponds to thenew “design” wavelength, which is at a different location than Θ=0. Byshifting the DBR-etalon reflection profile, a sharp transition mirrormay be designed. In the lossless case, the reflectance varies from 100%to 0% over δλ=(7.4×10⁻⁵)λ₀. A sharper transition can be obtained byincreasing |r_(g)|² and decreasing ΔL at which may, however, result in anarrower high reflectance bandwidth, more ripple, and more stringentfabrication tolerances.

Among the properties of the disclosed DBR-MRR are: for fixed values of αand τ, the maximum achievable reflectance is the same along a continuouscurve in the Θ-r plane; in the lossless case limit, 100% reflectance canbe obtained with weak reflective elements; the FP-MRR can generate aperiodic reflectance spectrum with peaks at the resonance wavelengths ofthe ring; the higher order grating DBR-MRR can generate a periodicreflectance spectrum with peaks at a select subset of the resonancewavelengths of the ring; the DBR-MRR with the grating occupying eitherhalf or all of the ring suppresses reflection at adjacent resonancewavelengths of the ring and thereby produces a single peak profile; and,the DBR-E-MRR reduces the FWHM of the DBR etalon and can be designed tofunction as either an narrow filter or a sharp transition mirror.

Integrating reflective elements into a ring resonator leads tostructures that are generally more compact than their separateequivalents, have virtually no side mode ripple, and offer control ofthe reflection at adjacent resonance wavelengths of the ring and therebyproduce a single peak profile. The DBR-E-MRR reduces the FWHM of the DBRetalon and can be designed to function as either a narrow filter or asharp transition mirror.

Achieving a high DBR-MRR reflectivity may need low propagation loss andcontrol of dimensions during fabrication, in particular the waveguidewidth and the waveguide bus-to-ring gap. A multi-parameter sensitivityanalysis of the half-ring DBR-MRR of FIG. 3 was performed using thegraphical approach. To get R_(max)>90%, the loss should be below 1.5dB/cm. For a fixed loss of 1 dB/cm. and R_(max)>90%, one can tolerate0.87<τ²<0.92: i.e., a gap of between about 34 and about 58 nm at 1550nm. This tolerance can be achieved using E-beam or nano-imprintlithography (NIL). The structure maintains a nearly constant FWHM forparameters in this range.

A number of fabrication techniques may be considered. E-beam lithographyis a low-throughput method, having high resolution and the flexibilityto modify device dimensions each time the device is fabricated.Nano-imprint lithography is a higher-throughput method with sub-10 nmresolution and excellent large area uniformity. An instrument such asthe Molecular Imprints (Austin, Tex.) Imprio 55 NIL may result in smoothsidewalls for high quality factor MRRs and accurate and repeatable τ andα values. During development, devices may be patterned so as to studythe effect of design parameters on the performance DBR-MRRs. Oncedesirable layouts are produced, NIL masks may be produced so as to massproduce devices. Although NIL masks may not have the flexibility tomodify the device dimensions without a new mask, the waveguide index orheight may be modified instead.

We have developed processes to deposit non-porous dielectric films ofamorphous Si (a-Si), SiNx and SiOxNy with arbitrary refractive indexfrom about 1.45 to about 3.3 and low loss at λ₀=1550 nm. The refractiveindex profile is shown in FIG. 17A. With the dimensions defined on themaster NIL mask, one can control waveguide properties, e.g., β and τ, byadjusting the optical mode shape using the core material composition andheight as control variables. The ability to control index permitsachieving specific design wavelengths. Since the index may becontinuously graded, post process trimming may be performed bydepositing a thin (<50 nm) blanket cladding of arbitrary index to adjustβ. FIG. 17B is a side view of a possible DBR-MRR structure. On Si waferswith a 6 μm thick thermal oxide layer (available from Rogue ValleyMicrodevices, Medford, Oreg.), a SiNx core with a constant or gradedcomposition could be deposited using plasma enhanced chemical vapordeposition (PECVD). The ridge for the bus and ring and the apodized DBRcould then be patterned with NIL and etched deep into the SiO₂ bottomcladding in a single self aligned processing step.

Table 1 shows nominal dimensions for a single-mode low-loss half-ringgrating (N=M). The table includes results from full 3D mode calculationswith PML in COMSOL (COMSOL Multiphysics, Burlington, Mass.), usingmeasured ellipsometry data, which includes refractive index dispersionand absorption. The choice of SiNx rather than Si as the core reducesabsorptive loss across the 900-1700 nm wavelength window. Note that Siand other semiconductor materials and various dielectric materials mayserve as the core layer or the cladding and are not intended to beexcluded by this specific example. Core thickness and index contrast toSiO₂ are parameters that may be controlled so as to keep the substrateleakage loss low.

TABLE 1 DBR-MRR parameters λ₀ n_(core) t h Ring/Bus 1.55 μm 2 400 nm 1μm Design Δw N w (apodization) DBR 200 1 μm 50 nm Design M g Avg. radiusRing 200 125 nm 30.6 μm Design n_(mode) Loss Q_(ring only) α_(r)Calculated 1.7 2 dB/cm 150000 ~1 κ² Δn |r_(DBR)|² R (λ₀) Calculated 0.274 × 10⁻³ 2.10% ~92%

Passive DBR-MRRs may serve as laser cavity mirrors when coupled to, orintegrated with, active devices. Two examples of such coupling are:quantum well intermixing (QWI) (FIG. 18A) and vertical coupling with ataper (see FIGS. 18B, C and D). QWI is a self aligned process soactive-passive coupling loss is negligible; however, the propagationloss depends on the quality of intermixing. Horizontally tapering thewaveguide to increase vertical mode size and achieve low-loss couplingto a purely passive waveguide may also be used.

In FIGS. 18B and C, the active and passive waveguides are made of asemiconductor material and are grown on top of each other whereby thepassive waveguide can be made to be passive by not including a quantumwell in its layer structure. In FIG. 18D, the passive waveguide is madeby depositing a dielectric material on top of the semiconductor gainregion. A thick bottom cladding may be used to reduce substrate leakageloss. Additional active-passive integration schemes may include, forexample, selective area regrowth or etching away the active region anddepositing passive dielectric for butt-joint coupling. InGaAs QWs may beused at 980 nm for the active material. Operation at other wavelengthsis also possible by using different semiconductor active regionmaterials and suitable substrates. See “Method of plasma etchingGa-based compound semiconductors,” (US PgPub 2010/0159706) which isincorporated herein by reference, which yields highly repeatable, verysmooth, anisotropic etch processes for Gallium based semiconductors athigh, low, and ultra-low etch rates (for example, 450, 17, and 2nm/min). The sidewall verticality was 90°±2° and etched surfaceroughness was comparable to as-grown epi-wafers, e.g. <0.3 nm RMS over a100 μm² area. The control of etched features is expected to be severalnanometers in all 3 dimensions. FIG. 19 shows preliminary results for anapodized grating on a piece of GaAs.

In an aspect, these concepts may be embodied in fiber optic componentsand devices using such components. For example a fiber Bragg grating FBG150 may be fusion spliced to a low loss 10%:90% optical coupler 110 withoptical fiber connections 130, as shown in FIG. 20. The resulting devicewould be the fiber-based equivalent of the previously described DBR-MRRwith κ²=0.1. Although the figure shows the splice 120 being formed at adiameter of the loop and the loop being circular, these features are notbe considered as limitations on the scope claimed herein.

Substituting a chirped FBG or long-period Bragg grating (LPG) or formingan etalon inside the fiber ring resonator would permit a variety ofdevices to be designed and constructed for purposes of, for example,chromatic dispersion compensation, pulse compression/stretching, orincreasing the sensitivity of existing linear FBG based temperature orstress sensors. Typical fiber based devices may have lower losses thanthe semiconductor/dielectric planar devices, and fiber devices may havea larger diameter ring thereby resulting in a narrower FSR and narrowerFWHM bandwidth. The half ring and full ring DBR-MRR configuration mayselect a single wavelength for reflection from the large number ofresonant modes which are a result of the larger diameter fiber ring.

The DBR-MRR half and full ring designs select a single wavelength forreflection when the DBR is a first order grating (i.e. Λ=λ₀/2n) and thegrating refractive index perturbation is sinusoidal. When the gratingrefractive index perturbation approximates a rectangular function, thegrating contains higher order harmonic frequencies and λ₀, λ₀/3, λ₀/5, .. . may also be reflected by the DBR-MRR; however, the higher orderharmonics are often so far outside the bandwidth of interest that theycan be ignored in a design, e.g. λ₀/3=500 nm for λ₀=1500 nm and mostlasers would only tune for a 100 nm bandwidth or less and thus notencounter the other resonance reflection wavelengths (see FIG. 21).

When a higher-order grating with a rectangular refractive indexperturbation is used, a comb of reflection peaks is obtained. In thiscircumstance, a subset of all of the microring resonances may experiencehigh reflection. In an example, 210=2×3×5×7 may be chosen as theazimuthal order so that a wide variety of grating orders can beobtained, e.g. 1^(st), 3^(rd), 5^(th), 7^(th), 15^(th), 21^(st),35^(th), and 105^(th), for both the full and half ring cases withoutchanging the device dimensions. The free spectral range (FSR) of themicroring without the grating is denoted as FSR₀. When the grating orderis N, the comb has a reflection every at every 420/N resonance. That is,for the 105^(th) order grating, the reflection peaks occur every 4^(th)resonance (the comb has FSR=4 FSR₀); for the 35^(th) order grating, thereflection peaks occur every 12^(th) resonance (comb FSR=12 FSR₀), . . .and for the 1^(st) order rectangular grating, they occur every 420^(th)resonance. Some examples are shown in FIG. 21. Other multiples of thecomb FSR may be obtained with other choices of the azimuthal order andthe grating order.

In yet another example, a DBR-MRR was, designed, fabricated and tested.A half-ring DBR was is created by modulating the width of the ringwaveguide. The device was designed for a 400 nm thick Si₃N₄ waveguidecore and SiO₂ bottom and top claddings. The bus and ring waveguidewidths were chosen to be 1 μm to achieve a single mode waveguide. Thering radius was determined using 2D finite elements method simulationwhich uses rotationally symmetric geometry of the ring resonator. At aresonant wavelength of 1550 nm for a quasi-transverse-electric (TE) modeof azimuthal order 200 (a mode with 200 field oscillations along thecircumference), the inner radius of the ring is set to 30.16 μm. The DBRwas realized by removing 50 nm×50 nm pieces from the inward side of thering waveguide, yielding a duty cycle of about 10%.

A small duty cycle was selected to get a low power reflectivity ofR=1.9% from the 200 DBR periods that occupy the top half of the ring.The resultant angular period of the grating is Φ₀=0.9°. Criticalcoupling requires bus to ring coupling coefficient of κ=0.5 whichcorresponds to a bus to ring gap of 125 nm.

To find the DBR reflectivity, a hybrid method that combines cylindricalcoordinates coupled mode formulation and 2D FEM simulations was used.The small indentations of the waveguide are treated as a perturbation tothe ring which couples the modes together. The electromagnetic fieldsinside the ring are approximated by a linear combination of the twounperturbed modes of the isolated ring with coefficients of the linearcombination being functions of the azimuthal angle φ (in plane anglefrom the center of the ring). These coefficients satisfy a system of twocoupled first order differential equations, which can be obtained usingthe fields of the resonance modes of the isolated ring and thegeometrical parameters of the indentation. The two differentialequations were solved to find the coefficients in the linearcombination. Using these solutions and the coupling matrix of the buscoupler, the total reflectivity of the ring is obtained.

The device was fabricated on a silicon wafer with a 3 μm thick grownSiO₂ layer. A 400 nm thick low pressure chemical vapor deposition(LPCVD) stoichiometric Si₃N₄ core was deposited on the grown oxide. Thedevice pattern was written using e-beam lithography and was transferredfrom the e-beam resist to the Si₃N₄ layer using reactive ion etching(RIE) with Freon gases. A 1.3 μm thick SiO₂ upper cladding was depositedusing plasma enhanced chemical vapor deposition (PECVD). After cleavingthe device, a 300 nm thick PECVD SiO₂ layer was deposited on thewaveguide facets as anti-reflection coating. FIG. 22B shows a scanningelectron microscope (SEM) image of the fabricated device prior to topcladding deposition. A close up view of the portion inside the smallrectangle shown in FIG. 22B is presented in FIG. 22C. The smallindentations on the inside of the ring waveguide can be seen in thisfigure and in FIG. 22D which is an angled view of the same structure.

Testing was performed by coupling polarized light from a tunable laserinto and out of the bus waveguide using optical tapered fibers. The buswaveguide width was tapered from 1 μm to 3 μm near the facet to increasethe fiber coupling efficiency. The reflected light was measured using afiber circulator.

Ring resonators with and without the DBR were tested. Plain rings had anunloaded Q of 1.56×10⁵. The reflection and transmission spectra of areflective ring are illustrated in FIG. 23. A reflection peak of −9 dBaccompanied by a transmission dip of −24.5 dB can be observed around1549.9 nm. This reflection peak is about 7.8 dB larger than the peaks atadjacent ring resonances. The reflectivity is suppressed at nearbyresonances because the DBR length is half the ring circumference, and asmentioned before, the nulls are located at the other ring resonancewavelengths. There is, however, some reflection due to ring waveguidesidewall roughness which causes smaller reflection peaks at these ringresonances.

FIG. 23B shows a zoomed in view of the transmission and reflectionspectra of the main resonance. In general, coupling of the modes causesseparation of the two transmission dips (resonance splitting). When thecoupling coefficient κ is greater than or equal to a critical couplingvalue, the splitting may be small enough that the overalltransmission/reflection spectrum shows a single dip/peak. The deepsingle transmission dip shown in FIG. 23B signifies the criticalcoupling of the ring. The observed asymmetry in the reflection peakspectrum is believed to be caused by interference of the light reflectedby the ring and by the first waveguide facet. The measured reflectionloss of −9 dB is believed to be mainly due to the fiber coupling anddecoupling losses. To account for the effects of reflection from thefacets and fiber coupling losses, simulated reflection and transmissionspectra of a model were fit to the measured data. The model includes thereflective ring and bus waveguide segments on each side of the ring.Coupling to the fibers is modeled by two two-port scattering matrices.The reflection and transmission spectra of the reflective ring werecalculated using a model which has been previously described.

The parameters of the ring resonator, DBR, facet reflectivities, andwaveguides propagation loss and lengths were chosen such that thesimulated reflection and transmission spectra both fit to those of themeasured device. The fitted results of the model were overlaid on themeasured results in FIG. 23B. The model response is in good agreementwith the measured results, allowing extraction of device performanceparameters. The bus waveguide to ring coupling coefficient of κ=0.52,waveguide loss of 2.3 dB/cm, and DBR peak power reflectivity of R=2.1%were found in this way. The DBR reflectivity is in good agreement withthe cylindrical coordinate coupled mode theory result which predictsR=1.9%. The slight difference is probably due to fabrication error. Thereflection and transmission spectra of the ring are also found usingthis method and are shown in FIG. 24. The device shows a maximumreflectivity of −0.34 dB corresponding to 92.3% power reflectivity. TheFWHM of the reflection peak is 0.4 nm. The simulated reflection andtransmission spectra of a conventional linear DBR with the same peakreflectivity and FWHM are also shown in FIG. 24.

The equivalent DBR length was 4.3 mm. The same waveguide propagationloss of 2.3 dB/cm was assumed in the linear DBR simulation. The about 1dB out of band transmission loss of the linear DBR is a result of thesubstantially greater length of the linear DBR.

The half-ring DBR-MRR is 70 times smaller in length, has a fasteramplitude roll-off with wavelength, and no side modes when compared tothe conventional DBR. The smooth narrow band high reflectivity peak at asingle wavelength and small device size make the half ring DBRattractive as an in-line mirror for low threshold narrow linewidth laserdiodes. The smaller footprint saves real estate and reduces tuning powercompared to the conventional DBRs. The methods disclosed herein may beprovided, at least in part, as a computer program product that may beembodied on a machine-readable medium having stored thereon instructionswhich may be used to cause a computer (or other electronic device) toperform the methods. For the purposes of this specification, the terms“machine-readable medium” shall be taken to include any medium that iscapable of storing or encoding a sequence of instructions or data forexecution by a computing machine or special-purpose hardware and thatcause the machine or special purpose hardware to perform any one of themethodologies or functions described. The term “machine-readable medium”shall accordingly be taken include, but not be limited to, solid-statememories, optical and magnetic disks, magnetic memories, and opticalmemories. Machine readable media are understood to be non-transientstorage devices.

Although the present invention has been explained by way of theembodiments described above, it should be understood to the ordinaryskilled person in the art that the invention is not limited to theembodiments, but rather that various changes or modifications thereofare possible without departing from the spirit of the invention.Accordingly, the scope of the invention shall be determined only by theappended claims and their equivalents.

What is claimed is:
 1. A device, comprising: an optical ring waveguidehaving a grating along at least a portion thereof, and an opticalwaveguide coupled to the optical ring waveguide, wherein at least one ofa reflection spectrum or a transmission spectrum of the device isdetermined by a circumference of the optical ring waveguide and at leastone of a reflection spectrum or a transmission spectrum of the grating.2. The device of claim 1, wherein the optical ring waveguide has anoptically active portion.
 3. The device of claim 1, wherein the opticalring waveguide is coupled to an active optical device.
 4. The device ofclaim 1, further comprising an active optical device inserted in theoptical ring waveguide.
 5. The device of claim 4, wherein the activeoptical device is a semiconductor device.
 6. The device of claim 3,wherein the active optical device is coupled to the optical ringwaveguide by insertion in the optical waveguide.
 7. The device of claim1, wherein the circumference of the ring is an integral multiple of arefractive index periodicity of the grating.
 8. The device of claim 1,further comprising an etalon inserted in the ring.
 9. The device ofclaim 8, wherein the etalon is a Fabry-Perot interferometer.
 10. Thedevice of claim 1, wherein the grating is formed by a plurality ofspaced regions having at least one of a higher or a lower refractiveindex when compared with the refractive index of the optical ringwaveguide.
 11. The device of claim 1, wherein the grating is formed by aplurality of perturbations spaced apart along a circumference of opticalring waveguide.
 12. The device of claim 11, where the spacing betweenthe perturbations is uniform.
 13. The device of claim 11, wherein thespacing between the perturbations is selected to result in a Bragggrating.
 14. The device of claim 11, wherein the spacing between theperturbations is selected to result in a chirped grating.
 15. The deviceof claim 1, wherein the optical ring waveguide is a circle shape. 16.The device of claim 1, wherein the optical ring waveguide is a racetrackshape.
 17. The device of claim 1, wherein a radius of curvature of theoptical ring waveguide is large with respect to a design wavelength. 18.The device of claim 1, wherein the optical ring waveguide has atransverse dimension suitable for single-mode optical transmission at adesign wavelength.
 19. The device of claim 1, wherein the gratingextends over half of the circumference of the optical ring waveguide.20. The device of claim 1, wherein the grating extends over thecircumference of the optical ring waveguide.
 21. An add-dropmultiplexer, comprising: an optical ring waveguide having a gratingalong at least a portion thereof; a first optical waveguide coupled tothe optical ring waveguide; and a second optical waveguide coupled tothe optical ring waveguide, wherein an add-drop wavelength is determinedby a diameter of the optical ring waveguide and a period of the grating.22. The add-drop multiplexer of claim 21, wherein an etalon is formed inthe optical ring waveguide.
 23. The add-drop multiplexer of claim 21,wherein the grating has a length equal to one of half of or the fulllength of the optical ring waveguide.